Band plus algebra preconditioners for two-level Toeplitz systems

In this paper we are interested in the fast and efficient solution of nm×nm symmetric positive definite ill-conditioned Block Toeplitz with Toeplitz Blocks (BTTB) systems of the form Tnm(f)x=b, where the generating function f is a priori known. The preconditioner that we propose and analyze is an extension of the one proposed in (D. Noutsos and P. Vassalos, Comput. Math. Appl., 56 (2008), pp. 1255–1270) and it arises as a product of a Block band Toeplitz matrix and matrices that may belong to any trigonometric matrix algebra. The underlying idea of the proposed scheme is to embody the well known advantages characterizing each component of the product when used alone. As a result we obtain spectral equivalence and a weak clustering of the eigenvalues of the preconditioned matrix around unity, ensuring the convergence of the Preconditioned Conjugate Gradient (PCG) method with a number of iterations independent of the partial dimensions. Finally, we compare our method with techniques already employed in the literature. A wide range of numerical experiments confirms the effectiveness of the proposed procedure and the adherence to the theoretical analysis.